Co-Moving Coordinates & Lapse Function N(t) in ADM Decomposition

[Tertius]

TL;DR Summary

The lapse function, or specifically $g_{00} = N(t)$ being a function of coordinate time, is often normalized away, ensuring proper time equals coordinate time.

In the ADM decomposition, like in the construction of the FRW metric, the coordinates are defined to be co-moving, so we know $$d\tau = dt$$ (i.e. the lapse function is normalized away)

Starting from a five-dimensional embedded hyperboloid (as in carroll pg. 324) $-u^2 + x^2 + y^2 + z^2 + w^2 = \alpha^2$, constructing the 4D deSitter coordinates with ${t, \chi, \theta, \phi}$ I calculate that $$g_{tt} = cosh[\frac{2t}{\alpha}]$$
Yet in the derivation, Carroll simply removes it without even a mention of where it went. The code I wrote to generate the metric correctly produces the other 3 non-zero components, so I assume there is not an error. This would have generated a cyclic dependence of proper time on coordinate time.

The lapse function, which specifically dictates the temporal "distance" between spatial sub-manifolds, is ultimately seen as non-dynamic. And though it specifically applies to ADM decomposed spacetimes, it is physically saying that you can always find "co-moving" coordinates for ADM spacetimes.
Further, I found this proof that in general you can always find a co-moving reference frame for any spacetime. (https://www.zweigmedia.com/diff_geom/Sec11.html).

My first question is: why is it always possible (if that's actually true) to find a co-moving reference frame?

Observers are defined as a smooth section of the manifold. Such that $g_{\mu\nu} = \eta_{ab} e^{a}_{\mu} e^{b}_{\nu}$ where $e$ are the basis vectors. So.. by definition, an observer (who views the local spacetime as Minkowskian) will always have a proper time that matches coordinate time.

Second question: Does this concept of observers always viewing flat space locally (the equivalence principle) contain part of the reason why it is always possible to find co-moving coordinates?

 

[anuttarasammyak]

Summary: The lapse function, or specifically $g_{00} = N(t)$ being a function of coordinate time, is often normalized away, ensuring proper time equals coordinate time.

My first question is: why is it always possible (if that's actually true) to find a co-moving reference frame?

If free fall elevators are arranged so that they will not collide and make networks with own coordinates numbers and own clock time attached to each of them, I assume they would make a co-moving reference frame. Though it is not a mathematical proof but a physical image.

 

[PAllen]

Summary: The lapse function, or specifically $g_{00} = N(t)$ being a function of coordinate time, is often normalized away, ensuring proper time equals coordinate time.
...
Further, I found this proof that in general you can always find a co-moving reference frame for any spacetime. (https://www.zweigmedia.com/diff_geom/Sec11.html).

This is a local proof, and its definition of comoving frame is explicitly local (meaning in some small region - see definition 11.2, "a locally inertial frame whose last basis vector is parallel to the curve"). There are more general proofs (e.g. in Wald), that coordinates with proper time = coordinate time (called, in general form, synchronous or Gaussian Normal) can be found in any spacetime, but they are 'rarely global'. FLRW is one of special cases where such a decomposition is global. Interestingly, it is possible to find such coordinates also for the total spacetime of a non-rotating perfectly spherically symmetrical collapse to a BH, covering all vacuum and matter regions of the complete manifold. Specifically the complete evolution of an Oppenheimer-Snyder collapse to a Schwarzschild BH, including all of the vacuum region that exists in this case, can all be covered in one Gaussian-Normal set of coordinates.

 

[PAllen]

Summary: The lapse function, or specifically $g_{00} = N(t)$ being a function of coordinate time, is often normalized away, ensuring proper time equals coordinate time.

In the ADM decomposition, like in the construction of the FRW metric, the coordinates are defined to be co-moving, so we know $$d\tau = dt$$ (i.e. the lapse function is normalized away)

In spacetimes in which it is possible to perform an ADM decomposition (not always possible - causality conditions are required), it is definitely not always possible to globally normalize the lapse function such that coordinate time equals proper time. This is possible only for a highly restricted subset of ADM decomposable spacetimes.

 

[PAllen]

To add a little detail, verging into 'A' level, see the following:

https://arxiv.org/abs/gr-qc/0703035

This covers, in section 3, the requirement of the strongest causality condition (global hyperbolicity) for the ADM formalism to work. In chapter 4 it discusses normalizing the lapse function to produce Gaussian Normal coordinates, explaining why this is always possible for some finite region around some chosen spacelike surface, and 'almost never' globally possible.

 

[PAllen]

Summary: The lapse function, or specifically $g_{00} = N(t)$ being a function of coordinate time, is often normalized away, ensuring proper time equals coordinate time.

It occurs to me that this is not correct per the reference in my last post. Instead you have (per eq. 4.43):

$g_{00} = -N^2 + \beta_i \beta^i$

in terms of the lapse and the shift vector. Of course the signs are a matter of convention. Thus, to get $g_{00}=1$ it is necessary to assume the shift vector is zero as well as normalizing the lapse. These are then the requirements for Gaussian Normal coordinates, which generally only can cover a finite patch of an ADM decomposable spacetime.

 

[Tertius]

To add a little detail, verging into 'A' level, see the following:

https://arxiv.org/abs/gr-qc/0703035

This link is excellent. I'm reading through the relevant chapters.

One quick question: Let's assume we have a non-unitary increasing g00(t), which is a function of coordinate time. Suppose an observer at point P starts their clock at time t1, and then later records the time at t2 and t3. Even if g00(t) is "lengthening" the proper time, if the observer records equally spaced proper time intervals, the underlying coordinate time would have a smaller interval, such that (t3-t2) < (t2-t1). But, it seems that there would be no way for the observer to know of a change in the underlying coordinate time. Is there a way the observer could see an effect from this?

 

[PAllen]

This link is excellent. I'm reading through the relevant chapters.

One quick question: Let's assume we have a non-unitary increasing g00(t), which is a function of coordinate time. Suppose an observer at point P starts their clock at time t1, and then later records the time at t2 and t3. Even if g00(t) is "lengthening" the proper time, if the observer records equally spaced proper time intervals, the underlying coordinate time would have a smaller interval, such that (t3-t2) < (t2-t1). But, it seems that there would be no way for the observer to know of a change in the underlying coordinate time. Is there a way the observer could see an effect from this?

NO, that is what distinguishes invariant physics (which is measurable) from pure coordinate features (coordinates used for description can be changed without affecting any predicted measurements).

 

[Tertius]

NO, that is what distinguishes invariant physics (which is measurable) from pure coordinate features (coordinates used for description can be changed without affecting any predicted measurements).

Thanks for the helpful responses. One last on-topic question: I've been reading a review paper about variable light speed theories, which got me thinking about g00(t). However, if the speed of light were dependent on coordinate time, wouldn't it then require invariance to be broken?

In Einstein's original GR paper, he talks about the coordinate speed of light around a massive object, and uses that to determine the amount of deflection light experiences. But then, as my original post mentions, the observer is defined as a smooth part of the manifold, so by very definition an observer in GR will always see c as a constant.

It seems like variable light speed theories have to accept that invariance breaking is inevitable.